1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
|
"""Betweenness centrality measures for subsets of nodes."""
from networkx.algorithms.centrality.betweenness import (
_add_edge_keys,
)
from networkx.algorithms.centrality.betweenness import (
_single_source_dijkstra_path_basic as dijkstra,
)
from networkx.algorithms.centrality.betweenness import (
_single_source_shortest_path_basic as shortest_path,
)
__all__ = [
"betweenness_centrality_subset",
"edge_betweenness_centrality_subset",
]
def betweenness_centrality_subset(G, sources, targets, normalized=False, weight=None):
r"""Compute betweenness centrality for a subset of nodes.
.. math::
c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\sigma(s, t)}
where $S$ is the set of sources, $T$ is the set of targets,
$\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
and $\sigma(s, t|v)$ is the number of those paths
passing through some node $v$ other than $s, t$.
If $s = t$, $\sigma(s, t) = 1$,
and if $v \in {s, t}$, $\sigma(s, t|v) = 0$ [2]_.
Parameters
----------
G : graph
A NetworkX graph.
sources: list of nodes
Nodes to use as sources for shortest paths in betweenness
targets: list of nodes
Nodes to use as targets for shortest paths in betweenness
normalized : bool, optional
If True the betweenness values are normalized by $2/((n-1)(n-2))$
for graphs, and $1/((n-1)(n-2))$ for directed graphs where $n$
is the number of nodes in G.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
Weights are used to calculate weighted shortest paths, so they are
interpreted as distances.
Returns
-------
nodes : dictionary
Dictionary of nodes with betweenness centrality as the value.
See Also
--------
edge_betweenness_centrality
load_centrality
Notes
-----
The basic algorithm is from [1]_.
For weighted graphs the edge weights must be greater than zero.
Zero edge weights can produce an infinite number of equal length
paths between pairs of nodes.
The normalization might seem a little strange but it is
designed to make betweenness_centrality(G) be the same as
betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
The total number of paths between source and target is counted
differently for directed and undirected graphs. Directed paths
are easy to count. Undirected paths are tricky: should a path
from "u" to "v" count as 1 undirected path or as 2 directed paths?
For betweenness_centrality we report the number of undirected
paths when G is undirected.
For betweenness_centrality_subset the reporting is different.
If the source and target subsets are the same, then we want
to count undirected paths. But if the source and target subsets
differ -- for example, if sources is {0} and targets is {1},
then we are only counting the paths in one direction. They are
undirected paths but we are counting them in a directed way.
To count them as undirected paths, each should count as half a path.
References
----------
.. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
Journal of Mathematical Sociology 25(2):163-177, 2001.
https://doi.org/10.1080/0022250X.2001.9990249
.. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
https://doi.org/10.1016/j.socnet.2007.11.001
"""
b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
for s in sources:
# single source shortest paths
if weight is None: # use BFS
S, P, sigma, _ = shortest_path(G, s)
else: # use Dijkstra's algorithm
S, P, sigma, _ = dijkstra(G, s, weight)
b = _accumulate_subset(b, S, P, sigma, s, targets)
b = _rescale(b, len(G), normalized=normalized, directed=G.is_directed())
return b
def edge_betweenness_centrality_subset(
G, sources, targets, normalized=False, weight=None
):
r"""Compute betweenness centrality for edges for a subset of nodes.
.. math::
c_B(v) =\sum_{s\in S,t \in T} \frac{\sigma(s, t|e)}{\sigma(s, t)}
where $S$ is the set of sources, $T$ is the set of targets,
$\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
and $\sigma(s, t|e)$ is the number of those paths
passing through edge $e$ [2]_.
Parameters
----------
G : graph
A networkx graph.
sources: list of nodes
Nodes to use as sources for shortest paths in betweenness
targets: list of nodes
Nodes to use as targets for shortest paths in betweenness
normalized : bool, optional
If True the betweenness values are normalized by `2/(n(n-1))`
for graphs, and `1/(n(n-1))` for directed graphs where `n`
is the number of nodes in G.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
Weights are used to calculate weighted shortest paths, so they are
interpreted as distances.
Returns
-------
edges : dictionary
Dictionary of edges with Betweenness centrality as the value.
See Also
--------
betweenness_centrality
edge_load
Notes
-----
The basic algorithm is from [1]_.
For weighted graphs the edge weights must be greater than zero.
Zero edge weights can produce an infinite number of equal length
paths between pairs of nodes.
The normalization might seem a little strange but it is the same
as in edge_betweenness_centrality() and is designed to make
edge_betweenness_centrality(G) be the same as
edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
References
----------
.. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
Journal of Mathematical Sociology 25(2):163-177, 2001.
https://doi.org/10.1080/0022250X.2001.9990249
.. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
https://doi.org/10.1016/j.socnet.2007.11.001
"""
b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
b.update(dict.fromkeys(G.edges(), 0.0)) # b[e] for e in G.edges()
for s in sources:
# single source shortest paths
if weight is None: # use BFS
S, P, sigma, _ = shortest_path(G, s)
else: # use Dijkstra's algorithm
S, P, sigma, _ = dijkstra(G, s, weight)
b = _accumulate_edges_subset(b, S, P, sigma, s, targets)
for n in G: # remove nodes to only return edges
del b[n]
b = _rescale_e(b, len(G), normalized=normalized, directed=G.is_directed())
if G.is_multigraph():
b = _add_edge_keys(G, b, weight=weight)
return b
def _accumulate_subset(betweenness, S, P, sigma, s, targets):
delta = dict.fromkeys(S, 0.0)
target_set = set(targets) - {s}
while S:
w = S.pop()
if w in target_set:
coeff = (delta[w] + 1.0) / sigma[w]
else:
coeff = delta[w] / sigma[w]
for v in P[w]:
delta[v] += sigma[v] * coeff
if w != s:
betweenness[w] += delta[w]
return betweenness
def _accumulate_edges_subset(betweenness, S, P, sigma, s, targets):
"""edge_betweenness_centrality_subset helper."""
delta = dict.fromkeys(S, 0)
target_set = set(targets)
while S:
w = S.pop()
for v in P[w]:
if w in target_set:
c = (sigma[v] / sigma[w]) * (1.0 + delta[w])
else:
c = delta[w] / len(P[w])
if (v, w) not in betweenness:
betweenness[(w, v)] += c
else:
betweenness[(v, w)] += c
delta[v] += c
if w != s:
betweenness[w] += delta[w]
return betweenness
def _rescale(betweenness, n, normalized, directed=False):
"""betweenness_centrality_subset helper."""
if normalized:
if n <= 2:
scale = None # no normalization b=0 for all nodes
else:
scale = 1.0 / ((n - 1) * (n - 2))
else: # rescale by 2 for undirected graphs
if not directed:
scale = 0.5
else:
scale = None
if scale is not None:
for v in betweenness:
betweenness[v] *= scale
return betweenness
def _rescale_e(betweenness, n, normalized, directed=False):
"""edge_betweenness_centrality_subset helper."""
if normalized:
if n <= 1:
scale = None # no normalization b=0 for all nodes
else:
scale = 1.0 / (n * (n - 1))
else: # rescale by 2 for undirected graphs
if not directed:
scale = 0.5
else:
scale = None
if scale is not None:
for v in betweenness:
betweenness[v] *= scale
return betweenness
|
